integral of x^2cos(x)-2xcos(x)

解答

\int x^{2} cos (x) - 2 x cos (x) d x

x^{2} sin (x) - 2 (- x cos (x) + sin (x)) - 2 (x sin (x) + cos (x)) + C

求解步骤

\int x^{2} cos (x) - 2 x cos (x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} cos (x) d x - \int 2 x cos (x) d x

\int x^{2} cos (x) d x = x^{2} sin (x) - 2 (- x cos (x) + sin (x))

\int 2 x cos (x) d x = 2 (x sin (x) + cos (x))

= x^{2} sin (x) - 2 (- x cos (x) + sin (x)) - 2 (x sin (x) + cos (x))

解答补常数

= x^{2} sin (x) - 2 (- x cos (x) + sin (x)) - 2 (x sin (x) + cos (x)) + C